Weighted estimates for bilinear fractional integral operators: a necessary and sufficient condition for power weights

Abstract

We consider weighted estimates for two bilinear fractional integral operators \(I_{2,\alpha }\) and \(BI_{\alpha }\). Moen (Collect Math 60:213–238, 2009) obtained a necessary and sufficient condition for \(I_{2,\alpha }\). However we know only some sufficient conditions for \(BI_{\alpha }\) which is a variant of the bilinear Hilbert transform. Restricted to power weights we obtain a necessary and sufficient condition for \(BI_{\alpha }\). We also prove a bilinear Stein–Weiss inequality.